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barotropic linear topographic wave, then it would radiate
energy forward along the slope as a shelf wave.
Cenedese and Linden
[2002] advanced this approach
by constructing an annular channel between radii of 13
and 43cm, with four axisymmetric topographic config-
urations: a flat bottom, a step, a raised shelf with a
slope, and a raised step with a slope. Buoyant fluid was
injected via an axisymmetric source at the inner wall of
the annulus, forming a symmetric coastal current that was
eventually subject to baroclinic instability. Introducing a
shelf produced instabilities with slightly higher azimuthal
wave numbers and resulted in secondary instability after
axisymmetry was reestablished.
Cenedese et al.
[2005] constructed a coastal geometry
qualitatively similar to that discussed in Section 12.2, but
in a rectilinear channel. An experimental slope of width
4cm and height 6cm separated a shelf from the deep
ocean, each of width 40 cm. The channel was 84cm long
and bounded at each end by vertical walls, one of which
was used to steer a baroclinic jet toward the slope. The
experimental results agreed qualitatively with the quasi-
geostrophic (QG) theory of
Carnevale et al.
[1999]: The
jet either split between the tank edge and the slope or
retroflected away from the slope.
Folkard and Davies
[2001] were the first to investigate
along-slope variations in the experimental topography, a
crucial element of the experiments discussed in Section
12.2. They employed a rectangular channel with a lin-
ear slope that was broken by a gap of varying length
filled with linearly stratified fluid. They then introduced a
gravity current at one end of the channel, hugging either
the slope or the “continental” channel wall. Breaking the
slope was found to slow and destabilize the current, lead-
ing in some situations to the formation of persistent eddies
at the upstream end of the gap.
Wolfe and Cenedese
[2006]
employed a very similar experimental configuration and
observed the same behavior in buoyant coastal currents
with a gap in the slope. They also introduced slopes of
various steepness in the gap and found that the current
was stabilized whenever its width was smaller than that of
the slope.
Sutherland and Cenedese
[2009] extended this approach
to a more realistically shaped trough in the experimental
shelf. This configuration is conceptually opposite to the
experiments described in Section 12.2, which represent a
headland projecting out from the continent. Their exper-
imental setup closely resembled that of
Whitehead and
Chapman
[1986]: A buoyant coastal current was injected
at one side of a cylindrical tank and then flowed around
its edge until it encountered a rectilinear slope of height
20cm and breadth
radius of curvature of the isobaths, in some cases forming
a counterrotating eddy in the canyon itself.
12.1.3. Large-Amplitude Shelf Waves
The theoretical and experimental studies of shelf waves
described above focus almost exclusively on linear dynam-
ics, for which a developed body of literature exists [e.g.,
Mysak
, 1980b]. Yet in experiments featuring coastal cur-
rents, the evolution is characterized by strongly nonlin-
ear features, such as the generation of eddies and large-
amplitude deviations of the streamlines away from the
isobaths [e.g.,
Sutherland and Cenedese
, 2009].
In fact, a description of coastal currents in terms of
nonlinear Rossby shelf waves is possible, but requires that
the shelf break is approximated as a discontinuity in depth
[
Longuet-Higgins
, 1968] and that the wave is long and
evolves slowly in time. Under these assumptions,
Haynes
et al.
[1993] derived a nonlinear wave equation based
on a solution of the QG equations in the limit of long
wavelength.
Clarke and Johnson
[1999] and
Johnson and
Clarke
[1999] obtained a dispersive correction to this wave
equation at next order in an asymptotic expansion based
upon a small ratio of wave amplitude to wave length (see
Section12.4.2).
Johnson and Clarke
[2001] summarized the
development and application of this theory.
This chapter focuses on experiments that are amenable
to exactly this kind of theoretical description. We con-
sider a channel similar to that of
Cenedese et al.
[2005],
with a narrow slope separating much broader shallow
and deep regions representing the continental shelf and
open ocean, respectively. This is illustrated schematically
in Figure 12.1. Our experiments elucidate the dynamics
of a retrograde coastal current flowing past a headland,
where the continental wall of the channel protrudes out,
narrowing the continental shelf. Examples of such a con-
figuration in the real ocean include the Agulhas current
in the Mozambique Channel [
Bryden et al.
, 2005;
Beal
et al.
, 2006, 2011] and the Gulf Stream at Cape Hatteras
[
Stommel
, 1972;
Johns and Watts
, 1986;
Pickart
, 1995].
In Section 12.2 we describe our experimental setup
and procedure and characterize the evolution of large-
amplitude waves generated by retrograde flow past a head-
land. In Section 12.3 we briefly review the QG equations
that underlie our nonlinear wave theory and numerical
simulations. In Section 12.4 we adapt nonlinear shelf wave
theory [
Johnson and Clarke
, 2001] to the annular channel,
and in Section 12.5 we describe our numerical scheme for
the QG equations. In Section 12.6 we compare the skill of
our theory, numerical solutions, and experiments in pre-
dicting the characteristics of large-amplitude wave break-
ing. Finally, in Section 12.7 we summarize our findings
and relate our results to previous studies of topographic
Rossby waves and coastal currents.
30cm. The slope was interrupted
for around 25cm by a canyon of length
∼
30cm. The
coastal current would separate from the bathymetry to
cross the canyon if the width of the flow exceeded the
∼
